1 ARCH 435 PROJECT MANAGEMENT Lecture 3: Project Time Planning (Arrow Diagramming Technique) Activity on Arrow (AOA)

1

ARCH 435

PROJECT MANAGEMENT

ARCH 435

PROJECT MANAGEMENT

Lecture 3: Project Time Planning (Arrow Diagramming Technique)

Activity on Arrow (AOA)

1. Each activity (task) is portrayed or presented by

an arrow.

2. The tail and head of the arrow denote the start

and finish of the activity whilst its duration is

shown in brackets below.

3. The length of the arrow has no significance nei-

ther has its orientation.

ARROW DIA-GRAM

Activity

] Duration]

Site Preparation

] 30]

4)As means of further defining the point in time, when an

activity starts or finishes, start and finish events are

added.

5)An event (= node = connector), unlike an activity,

does not consume time or resources, it merely repre-

sents a point in time at which something happens.

6)Numbers are given to the events to provide a unique

identity to each activity.

7)The first event in a project schedule is the start of the

project. The last event in a project schedule is the

end of the project.

ARROW DIA-GRAM

2010

Activity

[Duration]

Start Event Finish Event

Activity identification numbers

called event numbers

Activity Identifica-tion

i-j Numbers of Events

• The node at the tail of an arrow is the i-node.

• The node at the head of an arrow is the j-node.

i j

Activity

Duration

1) The network (the graphical representation of a project plan)

must have definite points of beginning and finish.

(The accuracy and usefulness of a network is dependent

mainly upon intimate knowledge of the project itself, and

upon the general qualities of judgment and skill of the

planning personnel.)

2) The arrows originate at the right side of a node and termi-

nate at the left side of a node.

3) Any two events may be directly connected by no more than

one activity.

4) Use symbols to indicate crossovers to avoid misunderstand-

ing.

Rules of Making Arrow Dia-gram

2010 30A B

Logical Relation-ships

• Node “20” is the j-node for activity “A” and it is also the i-node for activity “B”. Therefore, activity “A” is a predeces-sor to activity “B”.

• In other words activity “B” is a successor to activity “A”. • Activity B depends on activity A.

2010 40

30

50

Succeeding activities

• Event numbers must not be duplicated in a network.• j-node number is always greater than i-node number.

Logical Relation-ships

Concurrent activities (happens at the same time)

80

120

160

170

180

190

Logical Relation-ships

5) The network must be a logical representation of all the activities. Dummy Activities are used, where necessary for:

• Unique numbering, and• Logical sequencing.

Dummy activity is an arrow that represents merely a dependency of one activity upon another. A dummy activity has a zero time. It is also called dependency arrow.

Rules of Making Arrow Dia-gram

• The following network shows incorrect activity numbering.

90

100

110

50 70

60

80

A

B

Dummy Activi-ties

• For unique numbering, use a dummy activity.

90

100

110

50 70

60

80

A

B

75

Dummy Activi-ties

50

30

10

4020

For representing logical relationships,

you may need dummy activities.

Dummy Activi-ties

A

B

C

D

In this diagram:

Activity C depends on Activities A, B.

Activity D depends on Activities A, B.

LETS SAY, Activity C depends on Activity A ONLY, and Activity D depends on Activities A, B. How can we represent this relationship?

503010

4020

In this case, use a dummy activity to indicate the correct relationship.

Dummy Activi-ties

A

B

C

D

Now,

Activity C depends on Activities A ONLY.

Activity D depends on Activities A, B.

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6) There must be no "looping" in the network. The loop is an indication of faulty logic. The definition of one or more of the dependency relationships is not valid.

Rules of Making Arrow Dia-gram

13012010090

110

7) The network must be continuous (without unconnected activ-ities).

80

1006030

90502010 120

1107040

Rules of Making Arrow Dia-gram

8) Networks should have only one initial event and only one terminal event.

6030

90502010 120

1107040

Rules of Making Arrow Dia-gram

9) Before an activity may begin, all activities pre-ceding it must be completed (the logical rela-tionship between activities is (finish to start).

Rules of Making Arrow Dia-gram

Standard layout for recording data

Network Analysis (Compu-tation)

1. Occurrence times of Events = Early and late timings of event occurrence = Early and late event times

Earliest

Event

Time

Latest

Event

Time

Event

Label

Activity

TailHead

Activity

Early Event Time (EET = E =TE)

Forward Pass for Computing EETEach activity starts as soon as possible, i.e., as soon as all of its predecessor activities are completed.1. Direction: Left to right, from the beginning to the end of

the project2. Set: EET of the initial node = 03. Add: EETj = EETi + Dij

4. Take the maximumThe estimated project duration = EET of the last node.

Early Event Time (Earliest occurrence time for event) is the earliest time at which an event can occur, considering the duration of precedent activities.

ji

EETjEETi Activity

Dij

0

110 20

3

3

A B

830

4 C40

12

Early Event Times (EET = E =TE)

Early Event Times (TE)

12

4

0

80

4

5

9

K

L

M

50

1570

24

4

Early Event Times (TE) Ex-ample:

20

10

30

50

40

60

70

2

3

3

4

3

1

5

4

7

Early Event Times (TE)

20

10

0 30

50

40

3

60

70

2

3

3

4

3

1

5

4

79

16

82

4

Late Event Time (LET = L =TL)

Backward Pass for Computing LET1. Direction: Right to left, from the end to the beginning of

the project2. Set: LET of the last (terminal) node = EET .

3. Subtract: LETi = LETj - Dij

4. Take the minimum

Late Event Time (Latest occurrence time of event) is the latest time at which an event can occur, if the project is to be completed on schedule.

ji

LETi

EETjLETj

EETi Activity

Dij

Late Event Times (TL)

8

13

50

16

16

3

740

609

9

220

10

0 430

50

3

60

70

2

3

3

4

3

1

5

4

79

16

8

40

Late Event Times (TL), Ex-ample:

220

10

0 430

50

40

3

60

70

2

3

3

4

3

1

5

4

79

16

8

16

13

9

10

4

8

0

Late Event Times (TL), Ex-ample:

1. Early Start (ES): The earliest time at which an activ-ity can be started.

ESij = EETi

2. Early Finish (EF): The earliest time at which an ac-tivity can be completed.

EFij = ESij + Dij3. Late Finish (LF): The latest time at which an activity

can be completed without delaying project comple-tion.

LFij = LETj4. Late Start (LS): The latest time at which an activity

can be started.LSij = LFij Dij

2. Activity Times (Schedule)

Network Analysis (Compu-tation)

Example: Activity Times

220

10

0 430

50

40

3

60

70

2

3

3

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3

1

5

4

79

16

8

16

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4

8

0

ES20-50 = EET20 = 2EF20-50 = ES + D = 2 + 3 = 5LF20-50 = LET50 = 13LS20-50 = LF – D = 13 – 3 = 10

1. Total Float (TF)

• Total float or path float is the amount of time that an activity’s completion may be delayed without extending project completion time.

• Total float or path float is the amount of time that an activity’s completion may be delayed without affecting the earliest start of any activity on the network critical path.

Activity Floats

Network Analysis (Compu-tation)

1. Total Float (TF)

Total path float time for activity (i-j) is the total float associated with a path.

For arbitrary activity (ij), the total float can be written as:

Path Float =Total Float (TFij)= LSij ESij

= LFij EFij

= LETj – EETi Dij

Activity Floats

Network Analysis (Compu-tation)

Example: Total Float Times

TF20-50 = LS20-50 - ES20-50

TF20-50 = 10 – 2 = 8TF20-50 = LF20-50 - EF20-50

TF20-50 = 13 – 5 = 8TF20-50 = LET50 – EET20 - D20-50

TF20-50 = 13 – 2 – 3 = 8

220

10

0 430

50

40

3

60

70

2

3

3

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3

1

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4

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0

Network Analysis (Compu-tation)

2. Free Float (FF)

• Free float or activity float is the amount of time that an activity’s completion time may be delayed without affecting the earliest start of succeeding activ-ity.

• Activity float is “owned” by an individual activity, whereas path or total float is shared by all activities along a slack path.

• Total float always equals or exceeds free float (TF ≥ FF).

• For arbitrary activity (ij), the free float can be written as:Activity Float = Free Float (FFij)

= ESjk EFij = EETj – EETi Dij

Activity Floats

FF20-50 = ES50-70 – EF20-50

FF20-50 = 8 – 5 = 3FF20-50 = EET50 – EET20 - D20-50

FF20-50 = 8 – 2 – 3 = 3

Example: Free Float Times

220

10

0 430

50

40

3

60

70

2

3

3

4

3

1

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4

79

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Network Analysis (Compu-tation)

Interfering Float (ITF)

Interfering float is the difference between TF and FF.

If ITF of an activity is used, the start of some succeed-ing activities will be delayed beyond its ES.

In other words, if the activity uses its ITF, it “inter-feres” by this amount with the early times for the down path activity.

For arbitrary activity (ij), the Interfering float can be written as:

Interfering Float (ITFij)= TFij FFij = LETj EETj

3. Activity Floats

ITF20-50 = TF20-50 - FF20-50

IFF20-50 = 8 – 3 = 5ITF20-50 = LET50 – EET50

ITF20-50 = 13 – 8 = 5

Example: Interfering Float Times

220

10

0 430

50

40

3

60

70

2

3

3

4

3

1

5

4

79

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0

Network Analysis (Compu-tation)

Independent Float (IDF)

It is the amount of float which an activity will always possess no matter how early or late it or its predeces-sors and successors are.

The activity has this float “independent” of any slip-page of predecessors and any allowable start time of successors. Assuming all predecessors end as late as possible and successors start as early as possible.

IDF is “owned” by one activity.

In all cases, independent float is always less than or equal to free float (IDF ≤ FF).

3. Activity Floats

Network Analysis (Compu-tation)

Independent Float (IDF)

For arbitrary activity (ij), the Independent Float can be written as:Independent Float (IDFij)

= Max (0, EETj LETi – Dij)= Max (0, Min (ESjk) - Max

(LFli) Dij)

3. Activity Floats

Example: Independent Float Times

IDF20-50 = Max. (0, [EET50 – LET20 - D20-50])IDF20-50 = Max. (0, [8 – 10 – 3]) = 0

220

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